that our own conception of an infinite series is necessarily a conception of an indefinite and, therefore, of an essentially incomplete sequence, or else upon the assertion that an infinite collection, if viewed as real, would prove to be in itself of a quantitatively indefinite and changeable character. In the one case, the argument continues by showing that an indefinite and incomplete sequence is incapable of being taken to be a finished reality beyond our thought. In the other case, one insists that the quantitatively indefinite collection, if viewed as real, would stand in conflict with the very notion of reality, since the real is, as such, the determinate. “The essence of number,” says Mr. Bosanquet,[1] “is to construct a finite whole out of homogeneous units.” “An infinite number would be a number which is no particular number; for every particular number is finite.” “An infinite series[2] . . . is not anything which we can represent in the form of number, and therefore cannot be, quâ infinite series, a fact in our world. . . . Our constructive judgment requires parts and a whole to give it meaning. Parts unrelated to any whole cannot be judged real by our thought. Their significance is gone and they are parts of nothing.”
More detailed, in the application of the general charge of indefiniteness thus made against the conception of the infinite collections, are the often used arguments such as exemplify how, if infinite collections are possible at all, one infinite must be greater than another, while yet, as infinite and determinate, all the boundless collections must (so one supposes) be equal. Or, again, in a similar spirit, one has pointed out that, by virtue of the properties which we have deliberately attributed to the Ketten of the foregoing discussion, two infinite collections, if they existed, would be, in various senses of the term equal, at once equal and unequal to each other, or would
